Over low flame, heat a couple of tablespoons of the olive oil in a large skillet. Add the sliced potatoes and the garlic and season with salt. Cook and stir for about a minute, then add 3 tablespoons of water and cover the skillet. Cook, just until the potatoes are tender. Take off the cover and let any remaining water cook away. Set aside.
If you have a pizza stone, place it on a rack in the oven. Preheat oven to 475°F.
On a lightly floured surface, roll the dough quite thinly into a 13-inch round.
Transfer the pizza crust to a pizza peel or an upside-down pizza pan that you have sprinkled lightly with cornmeal. The cornmeal acts as ball bearings so the pizza can slide onto the stone. (see notes if you don’t have a stone or peel or if you have one and not the other)
Drizzle some olive oil on the crust and spread it around with a brush or your fingers. Spread about the pesto on next, followed by the red onion slices and most of the cheese. Now arrange the potatoes on top. They will stick together some, just spread them as evenly as you can. Sprinkle with salt and a big grind of black pepper. Sprinkle the remaining cheese evenly over the potatoes, and finish with the grated Parmesan.
Give the pizza peel a slight horizontal jiggle to make sure the crust will slide off the peel. Tip the peel to the back of the preheated baking stone, and pull it away quickly, releasing the crust onto the stone. Bake until the crust is brown, 8 to 10 minutes on the stone. (see notes)
Slide the baked pizza back onto the peel or a rimless baking pan, and then transfer to a serving platter.